Cramer-Rao bounds for M-PSK packets with random phase

We derive new Cramer-Rao bounds (CRBs) for the estimation of phase from a block of random M-PSK (M=8) symbols for the case where the phase to be estimated is a random variable (r.v.). Existing bounds for 2 and 4-PSK which model the phase as non-random are extended to obtain a new 8-PSK CRB. The new bound which models the phase as a r.v. is compared to the new and PSK bound which models the phase as non-random. With 8-PSK we see clearly that use of the random phase CRB more accurately models the behavior if the phase, as normally happens, is suppose to be constrained to the interval [-π/M, π/M)